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COT3100, Spring 2000
Assigned: 2/28/2000
S. Lang
Assignment #3 (40 pts.)
Due: 3/8 in class or on 3/9 or 3/10
during your lab class
First, we define some terms and notations.
Definition.
Consider a binary relation
R
⊂
A
×
B
.
The
inverse
of
R
, denoted
R
–1
, is a binary
relation
⊂
B
×
A
such that
R
–1
= {(
b
,
a
)  (
a
,
b
)
∈
R
}, that is,
R
–1
contains pairs of elements which
have the reverse order as they are in relation
R
.
(The text calls
R
–1
the
converse
of
R
, denoted
R
c
.)
Definition.
Let
R
⊂
A
×
A
denote a binary relation.
The following relations defined over
A
are
called
closures
:
(a)
The
reflexive closure
of
R
is
r
(
R
) =
R
∪
{(
a
,
a
) 
a
∈
A
}.
(b) The
symmetric closure
of
R
is
s
(
R
) =
R
∪
R
–1
.
(c)
The
transitive closure
of
R
is
t
(
R
) =
R
∪
R
2
∪
R
3
∪
...
, where
R
2
=
R
R
,
R
3
=
R
2
R
, etc.,
where
denotes relation composition.
Thus, (
a
,
b
)
∈
t
(
R
)
⇔
(
a
,
b
)
∈
R
n
,
for some
n
≥
1
⇔
there exist
a
1
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This document was uploaded on 07/14/2011.
 Spring '09

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